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There are several instances of Pure Type Systems (PTS) combining dependent types and subtyping in the literature. For example Subtyping dependent types David Aspinall, Adriana Compagnoni. TCS, 2001 http://www.sciencedirect.com/science/article/pii/S0304397500001754 Pure Type Systems with Subtyping Jan Zwanenburg. TLCA 1999 http://dx.doi.org/10.1007/3-540-48959-2_27 Unifying Typing and Subtyping Yanpeng Yang, Bruno C. d. S. Oliveira, OOPSLA 2017 http://i.cs.hku.hk/~bruno/papers/oopsla17.pdf The latter two feature F-omeag-sub-like higher-order subtyping, but none have polarized higher-order subtyping (in the style of Abel's 2008 paper mentioned by Gabriel). Cheers /Sandro On Tue, Dec 12, 2017 at 1:07 PM, Sandro Stucki <[email protected]> wrote: > There are several instances of Pure Type Systems (PTS) combining > dependent types and subtyping in the literature. For example > > Subtyping dependent types > David Aspinall, Adriana Compagnoni. TCS, 2001 > http://www.sciencedirect.com/science/article/pii/S0304397500001754 > > Pure Type Systems with Subtyping > Jan Zwanenburg. TLCA 1999 > http://dx.doi.org/10.1007/3-540-48959-2_27 > > Unifying Typing and Subtyping > Yanpeng Yang, Bruno C. d. S. Oliveira, OOPSLA 2017 > http://i.cs.hku.hk/~bruno/papers/oopsla17.pdf > > The latter two feature F-omeag-sub-like higher-order (HO) subtyping, > but none have polarized higher-order subtyping (in the style of Abel's > 2008 paper mentioned by Gabriel). > > Cheers > /Sandro > > On Tue, Dec 12, 2017 at 11:45 AM, Gabriel Scherer > <[email protected]> wrote: >> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ] >> >> There is a language with dependent types and subtyping (including >> contravariant functions) in: >> >> Normalization by Evaluation for Sized Dependent Types >> Andreas Abel, Andrea Vezzosi, and Theo Winterhalter (2017) >> http://www.cse.chalmers.se/~abela/icfp17-long.pdf >> >> However, subtyping there is not "higher-order" in the sense of having >> type-indexed parameters themselves indicate a variance (you cannot >> abstract over all covariant parametrized types) -- pi-types only track >> relevant and irrelevant abstraction. In contrast, see the >> "higher-order subtyping" for F-omega-sub in >> >> Polarized Subtyping for Sized Types >> Andreas Abel, 2008 >> http://www.cse.chalmers.se/~abela/mscs06.pdf >> >> >> For higher-order subtyping in dependent systems, the two references >> I know of happen to be also mentioned on the nLab wiki: >> >> https://ncatlab.org/nlab/show/directed+homotopy+type+theory >> >> they are the work by Harper and Licata on directed type theory (and >> Dan Licata's PhD thesis), and Andreas Nuyts' master thesis. >> >> 2-Dimensional directed dependent type theory >> Robert Harper, Dan Licata, 2011 >> http://www.cs.cmu.edu/~drl/pubs/lh102dtt/lh102dtt.pdf >> >> Dependently Typed Programming with Domain-Specific Logics >> Dan Licata, 2011 >> http://www.cs.cmu.edu/~drl/pubs/thesis/thesis.pdf >> >> Towards a Directed Homotopy Type Theory based on 4 Kinds of Variance >> Andreas Nuyts, 2015 >> http://people.cs.kuleuven.be/~dominique.devriese/ThesisAndreasNuyts.pdf >> >> >> On Tue, Dec 12, 2017 at 10:57 AM, Giacomo Bergami <[email protected] >>> wrote: >> >>> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list >>> ] >>> >>> Hello Everyone, >>> >>> I am trying to check if it is possible to do reflection (as in >>> Java) using "type safe" languages and, therefore, I am wondering if there >>> is a language having dependent types with subtyping (in particular, I'm not >>> talking of subtyping as in types' universes, but as in record subtyping). >>> All the infos I got was a paper by Luca Cardelli dated 2000/2001 but, since >>> then, it seems that whether the type system is consistent or not is still >>> an open problem ( http://lucacardelli.name/Papers/Dependent% >>> 20Typechecking.US.pdf ). >>> Therefore, I'm wondering if there are any advances on this >>> regard: moreover, it seems to be that no proof assistant supports this >>> technology. >>> Thanks in advance for any reply, >>> >>> Giacomo Bergami >>> Ph.D Student >>> University of Bologna >>> https://jackbergus.github.io/ >>>
